On a class of solutions to the generalized derivative Schrödinger equations II

Linares, Felipe; Ponce, Gustavo; Santos, Gleison N.

doi:10.1016/j.jde.2019.01.004

Cited by 14 publications

(8 citation statements)

References 45 publications

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“…and studied stability of nondegenerate solitons by applying the abstract theory of Grillakis, Shatah and Strauss [12,13] (see also [14] for partial results in this direction). Although well-posedness in the energy space for (gDNLS) was assumed in [27], the well-posedness problem was later dealt with in [41,18,26].…”

Section: Introductionmentioning

confidence: 99%

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Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

Fukaya

Hayashi

2020

Trans. Amer. Math. Soc.

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We consider the following nonlinear Schrödinger equation with derivative: (1) iut = −uxx − i|u| 2 ux − b|u| 4 u, (t, x) ∈ R × R, b ∈ R. If b = 0, this equation is a gauge equivalent form of the well-known derivative nonlinear Schrödinger (DNLS) equation. The equation (1) for b ≥ 0 has degenerate solitons whose momentum and energy are zero, and if b = 0, they are algebraic solitons. Inspired from the works [29, 8] on instability theory of the L 2-critical generalized KdV equation, we study the instability of degenerate solitons of (1) in a qualitative way, and when b > 0, we obtain a large set of initial data yielding the instability. The arguments except one step in our proof work for the case b = 0 in exactly the same way, and in particular the unstable directions of algebraic solitons are detected. This is a step towards understanding the dynamics around algebraic solitons of the DNLS equation. Contents

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Section: Introductionmentioning

confidence: 99%

“…direction). Although well-posedness in the energy space for (gDNLS) was assumed in [27], the well-posedness problem was later dealt with in [41,18,26].…”

Section: Introductionmentioning

confidence: 99%

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

Fukaya

Hayashi

2020

Trans. Amer. Math. Soc.

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“…These results were useful, via the pseudo-conformal transformation, to study the scattering problem for NLS with α ≥ 2/N close to the critical power α = 2/N . The highly-regular solutions were also used to prove local existence for the generalized derivative Schrödinger equation [26,27] and to the generalized Korteweg-de Vries equation [25]. We expect that the results in the present paper will be useful to derive similar results for the nonlinear Klein-Gordon equation (1.1).…”

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confidence: 65%

Local smooth solutions of the nonlinear Klein-gordon equation

Cazenave¹,

Naumkin²

2021

DCDS-S

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Given any µ 1 , µ 2 ∈ C and α > 0, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation ∂ttu − ∆u + µ 1 u = µ 2 |u| α u on R N , N ≥ 1, that do not vanish, i.e. |u(t, x)| > 0 for all x ∈ R N and all sufficiently small t. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from [Commun. Contemp. Math. 19 (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.

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“…In fact, this approach can be helpful in establishing well-posedness in equations with a potential that can be expressed as a Calderon-Zygmund operator. Inspired by the results in [6] (see also [5,7,14,16,17]), we introduce a class of initial data, which guarantees existence of local solutions of (1.1) for nonlinearity with power p < 2. Main difficulties arise in our analysis due to the presence of the Riesz potential operator and the lack of regularity of the term 1 |x| N −γ * |u| p |u| p−2 u when p < 2.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.2. It is worth to emphasize that the results for the equations studied in [5,6,7,14,16,17] deal with weights of arbitrary size only limited by lower bounds. Thus, in these references, it is more natural to consider weights with integer powers.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$

Arora¹,

Riaño²,

Roudenko³

2020

Preprint

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We investigate the well-posedness in the generalized Hartree equation iut + ∆u + (|x| −(N−γ) * |u| p )|u| p−2 u = 0, x ∈ R N , 0 < γ < N , for low powers of nonlinearity, p < 2. We establish the local wellposedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin [6]. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the L 2 -supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

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On a class of solutions to the generalized derivative Schrödinger equations II

Cited by 14 publications

References 45 publications

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

Local smooth solutions of the nonlinear Klein-gordon equation

Well-posedness in weighted spaces for the generalized Hartree equation with $p<2$

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